Abstract.

For n≥3 distinct points in the d-dimensional unit sphere Open image in new window there exists a Möbius transformation such that the barycenter of the transformed points is the origin. This Möbius transformation is unique up to post-composition by a rotation. We prove this lemma and apply it to prove the uniqueness part of a representation theorem for 3-dimensional polytopes as claimed by Ziegler (1995): For each polyhedral type there is a unique representative (up to isometry) with edges tangent to the unit sphere such that the origin is the barycenter of the points where the edges touch the sphere.

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Acknowledgments.

I would like to thank Alexander Bobenko and Günter Ziegler for making me familiar with the problem of finding unique representatives for polyhedral types, and Ulrich Pinkall, who has provided the essential insight for this solution.

Eppstein, D.: Hyperbolic geometry, Möbius transformations and geometric optimization. Lecture given at the “Introductory Workshop in Discrete and Computational Geometry” of the Mathematical Sciences Research Institute, August 2003. Available on streaming video form the MSRI at http://www.msri.org/.